Theorem mircgrextend 29155

Description: Link congruence over a pair of mirror points. cf tgcgrextend 28948. (Contributed by Thierry Arnoux, 4-Oct-2020.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirtrcgr.e	⊢ ∼ = (cgrG‘𝐺)
mirtrcgr.m	⊢ 𝑀 = (𝑆‘𝐵)
mirtrcgr.n	⊢ 𝑁 = (𝑆‘𝑌)
mirtrcgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirtrcgr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirtrcgr.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirtrcgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mircgrextend.1	⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))

Assertion

Ref	Expression
mircgrextend	⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Proof of Theorem mircgrextend

Step	Hyp	Ref	Expression
1		mirval.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. 2 ⊢ − = (dist‘𝐺)
3		mirval.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		mirtrcgr.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		mirtrcgr.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
8		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
9		mirtrcgr.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐵)
10	1, 2, 3, 7, 8, 4, 6, 9, 5	mircl 29134	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11		mirtrcgr.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
12		mirtrcgr.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13		mirtrcgr.n	. . 3 ⊢ 𝑁 = (𝑆‘𝑌)
14	1, 2, 3, 7, 8, 4, 12, 13, 11	mircl 29134	. 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃)
15	1, 2, 3, 7, 8, 4, 6, 9, 5	mirbtwn 29131	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴))
16	1, 2, 3, 4, 10, 6, 5, 15	tgbtwncom 28952	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴)))
17	1, 2, 3, 7, 8, 4, 12, 13, 11	mirbtwn 29131	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋))
18	1, 2, 3, 4, 14, 12, 11, 17	tgbtwncom 28952	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋)))
19		mircgrextend.1	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))
20	1, 2, 3, 4, 5, 6, 11, 12, 19	tgcgrcomlr 28943	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋))
21	1, 2, 3, 7, 8, 4, 6, 9, 5	mircgr 29130	. . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴))
22	1, 2, 3, 7, 8, 4, 12, 13, 11	mircgr 29130	. . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋))
23	20, 21, 22	3eqtr4d 2805	. 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋)))
24	1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23	tgcgrextend 28948	1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  distcds 17438  TarskiGcstrkg 28890  Itvcitv 28896  LineGclng 28897  cgrGccgrg 28974  pInvGcmir 29125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-trkgc 28911  df-trkgb 28912  df-trkgcb 28913  df-trkg 28916  df-mir 29126

This theorem is used by:  mirtrcgr  29156